Existing direct methods of phase evaluation permit routine molecular structure determination for most crystalline compounds having fewer than 100 independent nonhydrogen atoms. The goal of this project is to extend these methods to compounds containing 500 atoms or more. Substances in this size range include many biologically important materials such as oligonucleotides and peptide antibiotics and hormones. Certain of these methods will have applicability to macromolecules as well, either for ab initio phase determination or phase refinement. Several approaches to the development of improved phasing methods are being pursued simultaneously. In traditional direct methods, the process leading from experimentally measured X-ray diffraction intensities to the values of individual phases is mediated by certain linear combinations of phases, the structure invariants. Phases are evaluated in a sequential manner, and the entire process fails when incorrect invariant values are assumed. Algebraic techniques are being used to derive formulas which should permit an accurate estimation of the values of individual structure invariants and accurate identification of those invariants which are weak links in the phasing process. An alternative approach is to bypass sequential phase evaluation altogether by substituting a process in which all phases are assigned simultaneously. The phases are then refined to their correct values by minimizing a function which effectively exploits the information contained in the conditional probability distributions of the structure invariants. It is particularly noteworthy that this process has been tested successfully for known 28-atom and 84-atom structures using random N-atom models to give the initial phases. Finally, the theoretical basis for extracting the phase information contained in intensity data measured at two or more wavelengths, using anomalous dispersion, is being derived.